Gauge Invariant Equations Research Articles

A minimal relativistic model of gravitation within standard restrictions of the Classical Theory of Fields

The minimal relativistic model of gravitation on the basis of the gauge-invariant theory of the linear scalar massless field is suggested. The principle of the multiplicative inclusion of gravitational interaction, the requirements being that the simplicity and invariance of the theory under the allowed (gauge) transformation of potential Ф → Ф′ = Ф + const as the basis of the approach, is used. A system of gauge-invariant gravitational field and matter equations is obtained and an energy-momentum tensor with a positively defined density of the field energy is constructed. The exact solutions to equations for the central static field and for fields of spherically symmetric and plane gravitational waves in the free space and in the material media are obtained.

Wave equations arising from perturbations of Bianchi III manifolds

We perform a linear perturbation study of two instances of Bianchi III manifolds: dust and radiation. We take the metric approach and perturb the metric component by component. Once we construct gauge-invariant variables we work without fixing the gauge and use the perturbations as ingredients of the gauge-invariant equations that result when the field equations are enforced to first order. Exact solutions to gauge-invariant equations are extracted explicitly. When possible, we rewrite our solutions as wave equations, giving the evidence of gravitational radiation and sound waves. In total, we list four gravitational wave equations, three sound wave equations and one equation coupling a gravitational wave to a sound wave. These resonances are not at all independent and we point out some identities. Some late-time behaviour is discussed. The approach is wholly general; the resulting equations are physically relevant and informative.

Detecting a Lorentz-violating field in cosmology

We consider cosmology in the Einstein-\AE{}ther theory (the generally covariant theory of gravitation coupled to a dynamical timelike Lorentz-violating vector field) with a linear \AE{}-Lagrangian. The $3+1$ spacetime splitting approach is used to derive covariant and gauge invariant perturbation equations which are valid for a general class of Lagrangians. Restricting attention to the parameter space of these theories which is consistent with local gravity experiments, we show that there are tracking behaviors for the \AE{} field, both in the background cosmology and at the linear perturbation level. The primordial power spectrum of scalar perturbations in this model is shown to be the same as that predicted by standard general relativity. However, the power spectrum of tensor perturbation is different from that in general relativity, but has a smaller amplitude and so cannot be detected at present. We also study the implications for late-time cosmology and find that the evolution of photon and neutrino anisotropic stresses can source the \AE{} field perturbation during the radiation and matter dominated epochs, and as a result the CMB and matter power spectra are modified. However, these effects are degenerate with respect to other cosmological parameters, such as neutrino masses and the bias parameter in the observed galaxy spectrum.

Cosmology of Ricci-tensor-squared gravity in the Palatini variational approach

We consider the cosmology of the Ricci-tensor-squared gravity in the Palatini variational approach. The gravitational action of standard general relativity is modified by adding a function $f({R}^{ab}{R}_{ab})$ to the Einstein-Hilbert action, and the Palatini variation is used to derive the field equations. A general method of obtaining the background and first-order covariant and gauge-invariant perturbation equations is outlined. As an example, we consider the cosmological constraints on such theories arising from the supernova type Ia and cosmic microwave background observations. We find that the best fit to the data is a non-null leading-order correction to Einstein gravity, but the current data exhibit no significant preference over the concordance model. The growth of nonrelativistic matter density perturbations at late times is also analyzed, and we find that a scale-dependent (positive or negative) sound-speed-squared term generally appears in the growth equation for small-scale density perturbations. We also estimate the observational bound imposed by the matter power spectrum for the model with $f({\mathbf{R}}^{ab}{\mathbf{R}}_{ab})=\ensuremath{\alpha}({\mathbf{R}}^{ab}{\mathbf{R}}_{ab}{)}^{\ensuremath{\beta}}$ to be roughly $|\ensuremath{\beta}|\ensuremath{\lesssim}\mathcal{O}({10}^{\ensuremath{-}5})$ so long as the dark matter does not possess compensating anisotropic stresses.

Cosmology of modified Gauss-Bonnet gravity

We consider the cosmology where some function $f(G)$ of the Gauss-Bonnet term $G$ is added to the gravitational action to account for the late-time accelerating expansion of the universe. The covariant and gauge invariant perturbation equations are derived with a method which could also be applied to general $f(R,{R}^{ab}{R}_{ab},{R}^{abcd}{R}_{abcd})$ gravitational theories. It is pointed out that, despite their fourth-order character, such $f(G)$ gravity models generally cannot reproduce arbitrary background cosmic evolutions; for example, the standard $\ensuremath{\Lambda}\mathrm{CDM}$ paradigm with ${\ensuremath{\Omega}}_{\mathrm{DE}}=0.76$ cannot be realized in $f(G)$ gravity theories unless $f$ is a true cosmological constant because it imposes exclusionary constraints on the form of $f(G)$. We analyze the perturbation equations and find that, as in the $f(R)$ model, the stability of early-time perturbation growth puts some constraints on the functional form of $f(G)$, in this case ${\ensuremath{\partial}}^{2}f/\ensuremath{\partial}{G}^{2}l0$. Furthermore, the stability of small-scale perturbations also requires that $f$ not deviate significantly from a constant. These analyses are illustrated by numerically propagating the perturbation equations with a specific model reproducing a representative $\ensuremath{\Lambda}\mathrm{CDM}$ cosmic history. Our results show how the $f(G)$ models are highly constrained by cosmological data.

Constraints onf(R)cosmology in the Palatini formalism

In this work we derive the covariant and gauge invariant perturbation equations in general theories of $f(R)$ gravity in the Palatini formalism to linear order and calculate the cosmic microwave background (CMB) and matter power spectra for an extensively discussed model, $f(R)=R+\ensuremath{\alpha}(\ensuremath{-}R{)}^{\ensuremath{\beta}}$, which is a possible candidate for the late-time cosmic accelerating expansion found recently. These spectra are discussed and found to be sensitively dependent on the value of $\ensuremath{\beta}$. We are thus able to make stringent constraints on $\ensuremath{\beta}$ from cosmological data on CMB and matter power spectra: The three-year Wilkinson Microwave Anisotropy Probe (WMAP) data alone give a constraint $|\ensuremath{\beta}|\ensuremath{\lesssim}\mathcal{O}({10}^{\ensuremath{-}3})$ while the joint WMAP, Supernova Lagacy Survey (SNLS), and Sloan Digital Sky Survey (SDSS) data sets tighten this to $\ensuremath{\beta}\ensuremath{\sim}\mathcal{O}({10}^{\ensuremath{-}6})$, about an order of magnitude more stringent than the constraint from SDSS data alone, which makes this model practically indistinguishable from the standard $\ensuremath{\Lambda}\mathrm{CDM}$ paradigm.

Gauge invariant Lagrangian construction for massive higher spin fermionic fields

Abstract We formulate a general gauge invariant Lagrangian construction describing the dynamics of massive higher spin fermionic fields in arbitrary dimensions. Treating the conditions determining the irreducible representations of Poincare group with given spin as the operator constraints in auxiliary Fock space, we built the BRST charge for the model under consideration and find the gauge invariant equations of motion in terms of vectors and operators in the Fock space. It is shown that like in massless case [I.L. Buchbinder, V.A. Krykhtin, A. Pashnev, Nucl. Phys. B 711 (2005) 367, hep-th/0410215 ], the massive fermionic higher spin field models are the reducible gauge theories and the order of reducibility grows with the value of spin. In compare with all previous approaches, no off-shell constraints on the fields and the gauge parameters are imposed from the very beginning, all correct constraints emerge automatically as the consequences of the equations of motion. As an example, we derive a gauge invariant Lagrangian for massive spin 3/2 field.

On the necessity of non-Riemannian acoustic spacetime in fluids with vorticity

The necessity of a newly proposed [L.C. Garcia de Andrade, Phys. Rev. D 70 (2004) 64004] non-Riemannian acoustic spacetime structure called acoustic torsion of sound wave equation in fluids with vorticity are discussed. It is shown that this structure, although not always necessary is present in fluids with vorticity even when the perturbation is rotational. This can be done by solving the Bergliaffa et al. [Physica D 191 (2004) 121] gauge invariant equations for sound, superposed to a general background flow, needs to support a non-Riemannian acoustic geometry in effective spacetime. Bergliaffa et al. have previously shown that a Riemannian structure cannot be associated to this gauge invariant general system.

Small-signal ac response: a self-consistent non-equilibrium Green's function approach

Starting from the fundamental equations for non-equilibrium Green's functions (NEGFs) and taking the internal induced ac potentials into account by self-consistent iteration, this paper obtains the gauge-invariant equations of ac charge and ac current in terms of NEGFs. We proved the gauge invariance of our formulation and performed the displacement current partition. As regards the computation, we found that a severe error would arise from numerical integration and destroy the gauge invariance of the theory. Fortunately, it can be corrected by adding a term and the huge computational burden that would otherwise emerge is avoided. Finally, we computed the admittance dispersion of a resonant tunnelling diode as a preliminary application. The results show that the peaks of total admittance have a direct relation to the peaks in the density of states and the displacement current always plays the role of a retarding ac current.

The Infeld-van der Waerden formalisms for general relativity

The elementary aspects of the spin-affine geometry borne by the Infeld-van der Waerden formalisms for classical general relativity are considered. It is explicitly shown that the symmetric parts of any admissible spin connexions behave covariantly under the action of the Weyl gauge group. The construction of a set of generalized covariant commutators provides a complete description of the corresponding curvature structures. The relevant computational procedures lead to a natural curvature splitting for each formalism which generally involves only the sum of purely gravitational and electromagnetic contributions. Under certain circumstances, the implementation of all commutators gives rise to a system of gauge-invariant wave equations for geometric photons and gravitons which carry prescribed index configurations.

Scalar field and electromagnetic perturbations on locally rotationally symmetric spacetimes

We study scalar field and electromagnetic perturbations on locally rotationally symmetric (LRS) class II spacetimes, exploiting a recently developed covariant and gauge-invariant perturbation formalism. From the Klein–Gordon equation and Maxwell's equations, respectively, we derive covariant and gauge-invariant wave equations for the perturbation variables and thereby find the generalized Regge–Wheeler equations for these LRS class II spacetime perturbations. As illustrative examples, the results are discussed in detail for the Schwarzschild and Vaidya spacetime, and briefly for some classes of dust universes.

Gauge invariant wave equations in curved space-times and primordial magnetic fields

The inflationary production of magnetic field seeds for galaxies is discussed. The analysis is carried out by writing the wave equation of the electromagnetic field in curved space-times. The conformal invariance is broken by taking into account the interaction of the electromagnetic field with the curvature tensor of the form $\ensuremath{\lambda}{R}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}\ensuremath{\delta}}{F}^{\ensuremath{\alpha}\ensuremath{\beta}}{F}^{\ensuremath{\gamma}\ensuremath{\delta}}$. Such a term induces an amplification of the magnetic field during the reheating phase of the universe, but no growth of the magnetic field occurs in the de Sitter epoch. The resulting primordial magnetic field turns out to have strengths of astrophysical interest.

Perturbations in cosmologies with a scalar field and a perfect fluid

We study the properties of cosmological density perturbations in a multi-component system consisting of a scalar field and a perfect fluid. We discuss the number of degrees of freedom completely describing the system, introduce a full set of dynamical gauge-invariant equations in terms of the curvature and entropy perturbations, and display an efficient formulation of these equations as a first-order system linked by a fairly sparse matrix. Our formalism includes spatial gradients, extending previous formulations restricted to the large-scale limit, and fully accounts for the evolution of an isocurvature mode intrinsic to the scalar field. We then address the issue of the adiabatic condition, in particular demonstrating its preservation on large scales. Finally, we apply our formalism to the quintessence scenario and clearly underline the importance of initial conditions when considering late-time perturbations. In particular, we show that entropy perturbations can still be present when the quintessence field energy density becomes non-negligible.

Wave equation for sound in fluids with vorticity

We use Clebsch potentials and an action principle to derive a complete closed system of gauge-invariant equations for sound superposed on a general background flow. Our system reduces to the Unruh [Phys. Rev. Lett. 46 (1981) 1351] and Pierce [J. Acoust. Soc. Am. 87 (1990) 2292] wave equations when the flow is irrotational, or slowly varying. We illustrate our formalism by applying it to waves propagating in a uniformly rotating fluid where the sound modes hybridize with inertial waves.

The price of an exact, gauge-invariant RG-flow equation

We combine old ideas about exact renormalization-group-flow (RGF) equations with the Vilkovisky–De Witt (VDW) approach to reparametrization invariant effective actions and arrive at a new, exact, gauge-invariant RGF equation. The price to be paid for such a result is that both the action and the RGF equation depend explicitly upon the base point (in field space) needed for the VDW construction. We briefly discuss the complications originating from this fact and possible ways to overcome them.

Experimental string field theory

We develop efficient algorithms for level-truncation computations in open bosonic string field theory. We determine the classical action in the universal subspace to level (18,54) and apply this knowledge to numerical evaluations of the tachyon condensate string field. We obtain two main sets of results. First, we directly compute the solutions up to level L=18 by extremizing the level-truncated action. Second, we obtain predictions for the solutions for L > 18 from an extrapolation to higher levels of the functional form of the tachyon effective action. We find that the energy of the stable vacuum overshoots -1 (in units of the brane tension) at L=14, reaches a minimum E_min = -1.00063 at L ~ 28 and approaches with spectacular accuracy the predicted answer of -1 as L -> infinity. Our data are entirely consistent with the recent perturbative analysis of Taylor and strongly support the idea that level-truncation is a convergent approximation scheme. We also check systematically that our numerical solution, which obeys the Siegel gauge condition, actually satisfies the full gauge-invariant equations of motion. Finally we investigate the presence of analytic patterns in the coefficients of the tachyon string field, which we are able to reliably estimate in the L -> infinity limit.

Geometric second order field equations for general tensor gauge fields

Higher spin tensor gauge fields have natural gauge-invariant field equations written in terms of generalised curvatures, but these are typically of higher than second order in derivatives. We construct geometric second order field equations and actions for general higher spin boson fields, and first order ones for fermions, which are non-local but which become local on gauge-fixing, or on introducing auxiliary fields. This generalises the results of Francia and Sagnotti to all representations of the Lorentz group.

Lectures on the functional renormalization group method

AbstractThese introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum field theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained by a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed briefly as an alternative way to arrive at the renormalization group equation. The scaling laws and fixed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scaler theory. The flattening of the effective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED.

Energy and momentum balance equations: An approach to quantum transport in closed circuits

Using the Heisenberg equations of motion, we have derived a set of gauge-invariant quantum-mechanical energy and momentum balance equations governing the transport of charged particles in a closed electric circuit, Because the driving electric field needs not to be uniform along the circuit, the balance equations provide a suitable tool for the investigation of quantum transport in mesoscopic systems. As an example, we investigated the global and local properties of energy dissipation in a quantum wire circuit under the influence of a localized elastic scatterer, modeled by a Dirac-delta potential. In the low-temperature linear-response regime we obtain the Landauer formula for the resistance of the elastic-scattering barrier. The localized electric field originating from this barrier is obtained using a self-consistent solution of the Poisson equation.

LOOP VARIABLES AND GAUGE INVARIANCE IN (OPEN) BOSONIC STRING THEORY

We give a simplified and more complete description of the loop variable approach for writing down gauge-invariant equations of motion for the fields of the open string. A simple proof of gauge invariance to all orders is given. In terms of loop variables, the interacting equations look exactly like the free equations, but with a loop variable depending on an extra parameter, thus making it a band of finite width. The arguments for gauge invariance work exactly as in the free case. We show that these equations are Wilsonian RG equations with a finite worldsheet cutoff and that in the ir limit, equivalence with the Callan–Symanzik β-functions should ensure that they reproduce the on-shell scattering amplitudes in string theory. It is applied to the tachyon–photon system and the general arguments for gauge invariance can be easily checked to the order calculated. One can see that when there is a finite worldsheet cutoff in place, even the U(1) invariance of the equations for the photon, involves massive mode contributions. A field redefinition involving the tachyon is required to get the gauge transformations of the photon into the standard form.